In order to calibrate the tests under the null hypothesis it should be pointed out that both \(\mathcal {T}_1\)  and \(\mathcal {T}_2\)  do not follow a free asymptotic distribution. The empirical estimate of MMD is a one-sample V-statistics and hence asymptotic distribution is difficult to obtain due to the degeneracy of V-statistics, which incorporates a correlation structure for the complete paired observations \(\mathbf {X}^{\text{com}}\)  [1]}. To address this issue we propose a wild bootstrap procedure for the first \(n_1\)  observations, while the remaining \(n_2+n_3\)  observations can be properly handled by permutations methods, that can achieve an exact type I error control. For each \(b=1,\dots ,B\) , it proceeds as follows:
